Week 4: Clustering and Classification
The topics of this chapter - clustering and classification - are handy and visual tools of exploring statistical data. Clustering means that some points (or observations) of the data are in some sense closer to each other than some other points. In other words, the data points do not comprise a homogeneous sample, but instead, it is somehow clustered.
In general, the clustering methods try to find these clusters (or groups) from the data. One of the most typical clustering methods is called k-means clustering. Also hierarchical clustering methods quite popular, giving tree-like dendrograms as their main output.
As such, clusters are easy to find, but what might be the “right” number of clusters? It is not always clear. And how to give these clusters names and interpretations?
Based on a successful clustering, we may try to classify new observations to these clusters and hence validate the results of clustering. Another way is to use various forms of discriminant analysis, which operates with the (now) known clusters, asking: “what makes the difference(s) between these groups (clusters)?”
In the connection of these methods, we also discuss the topic of distance (or dissimilarity or similarity) measures. There are lots of other measures than just the ordinary Euclidean distance, although it is one of the most important ones. Several discrete and even binary measures exist and are widely used for different purposes in various disciplines.
4.0 Packages for clustering and classification
library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6 ✔ purrr 0.3.4
## ✔ tibble 3.1.8 ✔ dplyr 1.0.9
## ✔ tidyr 1.2.0 ✔ stringr 1.4.1
## ✔ readr 2.1.2 ✔ forcats 0.5.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(dplyr)
library(ggplot2)
library(MASS)##
## Attaching package: 'MASS'
##
## The following object is masked from 'package:dplyr':
##
## selectlibrary(corrplot)## corrplot 0.92 loaded4.1 loading data and Exploring the data
#Loading the Boston data from the MASS package
data("Boston")
# Exploring the structure and the dimensions of the data set
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14The data set “Boston” from MASS library presents geographical, demographic, structural, economic and cultural description of different suburbs in Boston and their implication on housing values within different suburbs. Data set contains 14 variables (factors) and 506 observation and most variables are numerical.
The variables in the data set which influences the housing prices are:
crim = per capita crime rate by town. zn = proportion of residential land zoned for lots over 25,000 sq.ft. indus = proportion of non-retail business acres per town. chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). nox = nitrogen oxides concentration (parts per 10 million). rm = average number of rooms per dwelling. age = proportion of owner-occupied units built prior to 1940. dis = weighted mean of distances to five Boston employment centres. rad = index of accessibility to radial highways. tax = full-value property-tax rate per $10,000. ptratio = pupil-teacher ratio by town. black = 1000(Bk - 0.63)^2 where BkBk is the proportion of blacks by town. lstat = lower status of the population (percent). medv = median value of owner-occupied homes in $1000s.
Data is sourced from Harrison, D. and Rubinfeld, D.L. (1978) Hedonic prices and the demand for clean air. J. Environ. Economics and Management 5, 81–102. Belsley D.A., Kuh, E. and Welsch, R.E. (1980) Regression Diagnostics. Identifying Influential Data and Sources of Collinearity. New York: Wiley.
For more information about the “Boston” data set follow the link https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/Boston.html
4.2 Graphical Overview of the data set
#overview
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00#Plotting
long_Boston <- pivot_longer(Boston, cols=everything(), names_to = 'variable', values_to = 'value')
p1 <- ggplot(data=long_Boston)
p1 + geom_histogram(mapping= aes(x=value)) + facet_wrap(~variable, scales="free")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
#Relationship between the variables
cor_matrix <- cor(Boston)
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax ptratio
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431 0.2899456
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018 0.3832476
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320 0.1889327
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783 -0.3555015
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559 0.2615150
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819 0.4647412
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000 0.4608530
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304 1.0000000
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341 0.3740443
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593 -0.5077867
## black lstat medv
## crim -0.38506394 0.4556215 -0.3883046
## zn 0.17552032 -0.4129946 0.3604453
## indus -0.35697654 0.6037997 -0.4837252
## chas 0.04878848 -0.0539293 0.1752602
## nox -0.38005064 0.5908789 -0.4273208
## rm 0.12806864 -0.6138083 0.6953599
## age -0.27353398 0.6023385 -0.3769546
## dis 0.29151167 -0.4969958 0.2499287
## rad -0.44441282 0.4886763 -0.3816262
## tax -0.44180801 0.5439934 -0.4685359
## ptratio -0.17738330 0.3740443 -0.5077867
## black 1.00000000 -0.3660869 0.3334608
## lstat -0.36608690 1.0000000 -0.7376627
## medv 0.33346082 -0.7376627 1.0000000corrplot(cor_matrix, method="circle")
#Let's try a mixed plot with correlation values (closer the values to 1, stronger the correlation)
corrplot.mixed(cor_matrix, lower = 'number', upper = 'circle',tl.pos ="lt", tl.col='black', tl.cex=0.6, number.cex = 0.5)
In the correlation matrix, Color blue represents positive and red
represents the negative correlation. The values closer to +1 and -1
implies a stronger positive and negative correlation respectively. The
threshold is represented by different shades of blue and black and
decimal points which is indexed on the right side of the plot.
Based on the plot we can see:
Positive correlation between some variables for example:
- Proportional of non-retail business acres per town (indus) and nitrogen oxides concentration in ppm (nox)
- Index of accessibility to radial highways (rad) and full-value property-tax rate per $10,000 (tax)
Negative correlation between some variables for example:
- Proportional of non-retail business acres per town (indus) and weighted mean of distances to five Boston employment centers (dis)
- Nitrogen oxides concentration in ppm (nox) and weighted mean of distances to five Boston employment centers (dis)
4.3 Standardizing the dataset
Lets beging the scaling exercise to standardize the data set. Since
the Boston data contains only numerical values, so we can use the
function scale() to standardize the whole dataset as
mentioned in the exercise 4 instruction.
We also have the following equation from exercise which basically gives us the idea on scaling i.e., subtract the column means from corresponding means and dividing with the standard deviation.
\[scaled(x) = \frac{x - mean(x)}{ sd(x)}\]
#Saving the scaled data to the object
boston_scaled <- scale(Boston)
#Lets see
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)Scaling the data set with various scales makes the analyses easier. Scale() transforms the data into a matrix and array so for further analysis we can change it back to the data frame
Let’s scale further by creating a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE)
# look at the table of the new factor crime
table(crime)## crime
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# let's see the new data set now !!
summary(boston_scaled)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 [-0.419,-0.411]:127
## 1st Qu.:-0.5989 (-0.411,-0.39] :126
## Median :-0.1449 (-0.39,0.00739]:126
## Mean : 0.0000 (0.00739,9.92] :127
## 3rd Qu.: 0.2683
## Max. : 2.9865Now as mention in the exercise set “Divide and conquer”, lets divide train and test sets: training -> 80% and testing -> 20%
# number of rows in the Boston data set
n <- nrow(boston_scaled)
n## [1] 506# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)4.4 Fitting the linear discriminant analysis
Let’s move on fitting LDA on the training set.
Our target variable: crime(categorical)
# linear discriminant analysis
# Other variables are designated (.)
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## 0.2475248 0.2425743 0.2648515 0.2450495
##
## Group means:
## zn indus chas nox rm
## [-0.419,-0.411] 0.89447554 -0.8958692 -0.15421606 -0.8856586 0.4200376
## (-0.411,-0.39] -0.07378209 -0.3153201 -0.03128211 -0.5402615 -0.1894835
## (-0.39,0.00739] -0.37343545 0.2036824 0.27960087 0.3978276 0.1189162
## (0.00739,9.92] -0.48724019 1.0171737 -0.15302300 1.0420418 -0.4325790
## age dis rad tax ptratio
## [-0.419,-0.411] -0.8914002 0.9357258 -0.7004968 -0.6996510 -0.45106613
## (-0.411,-0.39] -0.2556164 0.3769261 -0.5377192 -0.4763222 -0.07796912
## (-0.39,0.00739] 0.4301848 -0.3891695 -0.3936844 -0.2751204 -0.33430767
## (0.00739,9.92] 0.7949744 -0.8417167 1.6375616 1.5136504 0.78011702
## black lstat medv
## [-0.419,-0.411] 0.36979482 -0.750037404 0.49756898
## (-0.411,-0.39] 0.31566531 -0.065445306 -0.03607504
## (-0.39,0.00739] 0.09089154 0.007096295 0.19478879
## (0.00739,9.92] -0.86757833 0.978459544 -0.71986628
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.14031540 0.598096615 -0.92383757
## indus -0.02930192 -0.189172614 0.22826688
## chas -0.10117097 -0.142233787 0.03149598
## nox 0.41573233 -0.794904909 -1.25110968
## rm -0.11314539 -0.023087098 -0.22031401
## age 0.33303790 -0.386946830 0.10834464
## dis -0.08707448 -0.225670182 0.26125036
## rad 3.05322455 0.876145452 0.07643080
## tax -0.04818628 0.024343114 0.23153680
## ptratio 0.13124250 -0.007118077 -0.25133969
## black -0.14400936 -0.003522677 0.13875654
## lstat 0.21595687 -0.113343967 0.39891878
## medv 0.18585476 -0.356563842 -0.20147563
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9480 0.0404 0.0116There are three discriminant function LD(proportion of trace) as follow:
LD1 LD2 LD3 0.9489 0.0362 0.0150
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
4.5 Predicting the LDA model
library(MASS)
ind <- sample(nrow(boston_scaled), size = nrow(boston_scaled) * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]
correct_classes <- test$crime
test <- dplyr::select(test, -crime)
lda.fit = lda(crime ~ ., data=train)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct [-0.419,-0.411] (-0.411,-0.39] (-0.39,0.00739] (0.00739,9.92]
## [-0.419,-0.411] 23 9 1 0
## (-0.411,-0.39] 2 17 2 0
## (-0.39,0.00739] 1 10 14 3
## (0.00739,9.92] 0 0 0 204.6 Model Optimization and K means clustering
Let’s work with clustering now Let’s calculate also the distances between observations to assess the similarity between data points. As instructed we calculated two distance matrix euclidean and manhattan.
#start by reloading the Boston data set
data("Boston")
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865#class of Boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame for futher analysis
boston_scaled <- as.data.frame(boston_scaled)# Calculating distances between the observation
# euclidean distance matrix
dist_eu <- dist(boston_scaled, method = "euclidean")
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# Manhattan distance matrix
dist_man <- dist(boston_scaled, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618Lets do K-means clustering (first with 4 clusters and then 3)
set.seed(123) #function set.seed() is used here to deal with the random assigning of the initial cluster centers when conducting k-means clustering
# k-means clustering
km <- kmeans(boston_scaled, centers = 4)
# plotting the scaled Boston data set with clusters
pairs(boston_scaled, col = km$cluster)
With 4 clusters, it appears that there is some overlapping between
clusters. lets try 3 clusters and check
# k-means clustering
km <- kmeans(boston_scaled, centers = 3)
# plotting the scaled Boston data set with clusters
pairs(boston_scaled, col = km$cluster)
with 3 clusters it is even worse !! We can try to determine the optimal
number of cluster for our model and see how it works
4.7 K-means: determine the k
#Investigating the optimal number of clusters
set.seed(123) #setseed function again
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line', ylab = "TWCSS", xlab = "Number of cluster")
From the plot it looks like the optimal number of clusters for our model is 2-3 as there is a sharp drop in TWCSS values. We can try with 2 because 2 seems more optimal as by 2.5 the drop is significant and we can see the substantial change.
# k-means clustering
km <- kmeans(boston_scaled, centers = 2)
# plot the Boston data set with clusters predicted
pairs(boston_scaled, col = km$cluster)
pairs(boston_scaled[6:10], col = km$cluster)
With 2 clusters, the model seem better, there is good separation for
example rad and tax. rm and age seems ok as well.